Semiclassical Dynamics with Exponentially Small Error Estimates

Abstract

We construct approximate solutions to the time--dependent Schrödinger equation $i \hbar (\partial ψ)/(\partial t) = - (\hbar^2)/2 Δψ+ V ψ$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-γ/\hbar}$, for some $C$ and $γ>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-γ'/\hbar^σ}$, for some $C', γ'>0$, and $σ>0$.